DOCSLIB.ORG

Quantum Finite-Time Thermodynamics: Insight from a Single Qubit Engine

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Finite-Time Thermodynamics: Insight from a Single Qubit Engine entropy Article Quantum Finite-Time Thermodynamics: Insight from a Single Qubit Engine Roie Dann 1,*, Ronnie Kosloff 1 and Peter Salamon 2 1 The Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel; [email protected] 2 Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720, USA; [email protected] * Correspondence: [email protected] Received: 6 September 2020; Accepted: 1 November 2020; Published: 4 November 2020 Abstract: Incorporating time into thermodynamics allows for addressing the tradeoff between efficiency and power. A qubit engine serves as a toy model in order to study this tradeoff from first principles, based on the quantum theory of open systems. We study the quantum origin of irreversibility, originating from heat transport, quantum friction, and thermalization in the presence of external driving. We construct various finite-time engine cycles that are based on the Otto and Carnot templates. Our analysis highlights the role of coherence and the quantum origin of entropy production. Keywords: finite-time thermodynamics; quantum thermodynamics; quantum heat engine; carnot cycle; otto cycle 1. Introduction The tradeoff between power and efficiency is well embedded in our everyday experience. It is witnessed in the performance of any realistic engine or refrigerator, from the operation of large nuclear plants, through the internal combustion engines of our automobiles, and all the way to microscopic biological engines and the quantum regime. Despite the intuitive notion, a theoretical analysis is quite involved, as it requires a theoretical construction that encompasses both thermodynamics and transient dynamics. The limiting case was first treated by Carnot, who linked an engine’s maximum attainable work production to reversible thermodynamic transformations, thereby obtaining the thermodynamic temperature scale and the universal optimal efficiency that only depends on the hot and cold bath temperatures [1]. Unlike efficiency, power requires knowledge of the transient dynamics, which is outside the realm of classical thermodynamics. Finite-time thermodynamics (FTT) was developed in order to include the limitations the process duration places on the performance of an engine [2–6]. Originally, the pioneers of FTT incorporated empirical kinetic laws to introduce an intrinsic timescale in the analysis of engine cycles [7,8]. Some results from these efforts are recapped in Section 2.1. In this paper, we address the need for kinetic laws by following a different approach: building upon a complete quantum description of the engine and baths. However, such complete quantum description is not as straightforward as it sounds. Quantum mechanics is a dynamical theory that can supply equations of motion for thermodynamic processes. The well established portion of this theory has predominantly dealt with closed systems that conserve entropy and, thus, cannot deal with dissipation phenomena at the heart of thermodynamic analyses. This forces us to turn to open quantum systems, whose description from first principles relies on a reduction from a closed composite system. The reduced description is achieved by tracing over the Entropy 2020, 22, 1255; doi:10.3390/e22111255 www.mdpi.com/journal/entropy Entropy 2020, 22, 1255 2 of 47 degrees of freedom of the surroundings, interacting with our system of interest. This description does not conserve entropy and allows for the exploration of thermodynamic processes in the quantum regime. Notably, the approach is based on the completely positive trace preserving (CPTP) dynamical map [9] and the Markovian Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equation [10,11]. A thermodynamically consistent dynamical framework [12] is obtained by a first principle derivation based on weak interaction of an open system with a heat bath. The derivation is commonly termed the ‘Davies construction’ [13]. The quantum open system approach resulted in a number of surprises, initially reported as claimed contradictions to the second law [14–19]. These include reported claims of breakdown of the Carnot bound in such engines [20–22]. In turn, these results led to resolutions, explained by unexpected work that is available from quantum resources, including coherence [23], squeezed bath [24], entanglement [25], and information [26,27]. Another set of surprises came from attempts to use a naive GKLS formalism with a time-dependent driving, which possibly violates the second law [28–31]. This led to the realization that the inconsistency arises from the derivation of the reduced dynamics of the system. For periodic driving, a thermodynamically consistent GKLS equation was derived in Ref. [32]. In the case of a general (non-periodic) driving, only the adiabatic master equation was available [33]. This fact did not limit the analysis of the Otto cycle model, which could be studied up to the limit of vanishing cycle times. On the contrary, the analysis of the Carnot cycle model was limited to the linear response regime. Only recently has a derivation of the GKLS master equation for rapid non-adiabatic driving become available [34], and it is this discovery whose implications we explore in the present paper. We adopt the dogma that thermodynamics and quantum mechanics address the same subject matter, therefore have to be consistent [35]. In this framework, quantum mechanics provides the tools for describing the dynamics, while the strict laws of thermodynamics must be obeyed. In addition, recent progress in the theory of quantum speed limits can illuminate fundamental bounds on the process timescale [36–41]. Engines have been an intrinsic part in the development of classical thermodynamics. Their analysis still serves as an integral part of current research in finite-time and quantum thermodynamics. These theories allow for describing engines more realistically including non-ideal performance. Any practical engine operates in a non-ideal irreversible mode. Typically, there are four sources of irreversible phenomena in engines: 1. Finite heat transport. 2. Friction. 3. Heat leaks. 4. Cost of switching contacts between subsystems. Following the thermodynamic tradition of learning from example, we employ the most elementary working medium, a spin one half system to explore a quantum version of finite-time thermodynamics. A decade ago, such an example would have been criticized as a theoretician’s toy with no connection to the world of real engines. The finite-time Otto type cycle, which our cars operate by, do not seem to be related to a single spin quantum engine. Nevertheless, recent experimental progress has enabled a realization of an Otto cycle engine constructed from a single spin of an atom in an ion trap [42], a single qubit in an impurity electron spin [43], or a working medium of ultracold hyperfine structure of Cs in a Rb bath [44]. The present paper begins by laying the quantum thermodynamic foundations for the qubit, giving the quantum definitions for energy, entropy, and temperature, Section 2.2. We continue by discussing sources of irreversibility: heat transport, Section3, the quantum origin of friction, Section4, and thermalization processes which combine heat transport and external work, Section5. The quantum version of finite-time thermodynamics is studied by constructing two basic engine platforms: Carnot and Otto. These models illuminate different aspects of the tradeoff between power and efficiency and the role of coherence on the engines performance, Sections6–8. Entropy 2020, 22, 1255 3 of 47 2. Some Preliminaries The unfortunate collision of the different usages of the word adiabatic in thermodynamics and quantum mechanics have been sidestepped by using the term “unitary dynamics” for dynamics along what thermodynamics would call an adiabat leaving the use of adiabatic for the quantum meaning. 2.1. Classical Engines Operating in Finite-Time Classical textbook treatments of heat engines define various kinds of engine cycles. These cycles are mostly four-stroke and consist of two unitary strokes and two open strokes in contact with a heat bath—one hot and one cold. Finite-time thermodynamic analyses of these cycles has given us the simplifying model of endoreversible processes—processes in which the participating systems are at each instant in equilibrium states and all irreversibility resides in the interactions between such systems. Endoreversible cycles play an important role by edging closer towards real cycles, being relatively easy to analyze and providing checks along the way for more ambitious treatments. They also provide an accurate picture of reality when the slow timescale is the interaction. The simplifying condition of instantaneous lossless adiabatic jumps, made possible for quantum systems using shortcuts to adiabaticity (cf. Section 4.3), is a hallmark simplifying feature that we inherit from these studies. Additionally, important for these analyses is a much older result known as the Gouy—Stodola theorem [45–48] which established a connection between dissipated work and entropy production, cf. U U DA = −T0DS (1) where the superscript U refers to the universe (all participating systems), S is the entropy, A is the available work, and T0 is the temperature at which heat is freely available which means it carries no available work. The environment temperature T0 is also used in the availability (also called exergy) state function A = E − T0S, where E is the internal energy. As a consequence,
Load more

Recommended publications
  • Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations
    Article Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations Jen-Tsung Hsiang 1,*,† and Bei-Lok Hu 2,† ID 1 Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China 2 Maryland Center for Fundamental Physics and Joint Quantum Institute, University of Maryland, College Park, MD 20742-4111, USA; [email protected] * Correspondence: [email protected]; Tel.: +86-21-3124-3754 † These authors contributed equally to this work. Received: 26 April 2018; Accepted: 30 May 2018; Published: 31 May 2018 Abstract: Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir.
    [Show full text]
  • The Quantum Thermodynamics Revolution
    I N F O R M A T I O N T H E O R Y The Quantum Thermodynamics Revolution By N A T A L I E W O L C H O V E R May 2, 2017 As physicists extend the 19th­century laws of thermodynamics to the quantum realm, they’re rewriting the relationships among energy, entropy and information. 48 Ricardo Bessa for Quanta Magazine In his 1824 book, Reflections on the Motive Power of Fire, the 28- year-old French engineer Sadi Carnot worked out a formula for how efficiently steam engines can convert heat — now known to be a random, diffuse kind of energy — into work, an orderly kind of energy that might push a piston or turn a wheel. To Carnot’s surprise, he discovered that a perfect engine’s efficiency depends only on the difference in temperature between the engine’s heat source (typically a fire) and its heat sink (typically the outside air). Work is a byproduct, Carnot realized, of heat naturally passing to a colder body from a warmer one. Carnot died of cholera eight years later, before he could see his efficiency formula develop over the 19th century into the theory of thermodynamics: a set of universal laws dictating the interplay among temperature, heat, work, energy and entropy — a measure of energy’s incessant spreading from more- to less-energetic bodies. The laws of thermodynamics apply not only to steam engines but also to everything else: the sun, black holes, living beings and the entire universe. The theory is so simple and general that Albert Einstein deemed it likely to “never be overthrown.” Yet since the beginning, thermodynamics has held a singularly strange status among the theories of nature.
    [Show full text]
  • Quantum Thermodynamics and Quantum Coherence Engines
    Turkish Journal of Physics Turk J Phys (2020) 44: 404 – 436 http://journals.tubitak.gov.tr/physics/ © TÜBİTAK Review Article doi:10.3906/fiz-2009-12 Quantum thermodynamics and quantum coherence engines Aslı TUNCER∗, Özgür E. MÜSTECAPLIOĞLU Department of Physics, College of Sciences, Koç University, İstanbul, Turkey Received: 17.09.2020 • Accepted/Published Online: 20.10.2020 • Final Version: 30.10.2020 Abstract: The advantages of quantum effects in several technologies, such as computation and communication, have already been well appreciated. Some devices, such as quantum computers and communication links, exhibiting superiority to their classical counterparts, have been demonstrated. The close relationship between information and energy motivates us to explore if similar quantum benefits can be found in energy technologies. Investigation of performance limits fora broader class of information-energy machines is the subject of the rapidly emerging field of quantum thermodynamics. Extension of classical thermodynamical laws to the quantum realm is far from trivial. This short review presents some of the recent efforts in this fundamental direction. It focuses on quantum heat engines and their efficiency boundswhen harnessing energy from nonthermal resources, specifically those containing quantum coherence and correlations. Key words: Quantum thermodynamics, quantum coherence, quantum information, quantum heat engines 1. Introduction Nowadays, we enjoy an unprecedented degree of control on physical systems at the quantum level. Quantum technologies promise exciting developments of machines, such as quantum computers, which can be supreme to their classical counterparts [1]. Learning from the history of industrial revolutions (IRs) associated with game- changing inventions such as steam engines, lasers, and computers, the natural question is the fundamental limits of those promised machines of quantum future.
    [Show full text]
  • Quantum Thermodynamics of Single Particle Systems Md
    www.nature.com/scientificreports OPEN Quantum thermodynamics of single particle systems Md. Manirul Ali, Wei‑Ming Huang & Wei‑Min Zhang* Thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. In this paper, we develop a theory of quantum thermodynamics that is applicable for arbitrary small systems, even for single particle systems coupled with a reservoir. We generalize the concept of temperature beyond equilibrium that depends on the detailed dynamics of quantum states. We apply the theory to a cavity system and a two‑level system interacting with a reservoir, respectively. The results unravels (1) the emergence of thermodynamics naturally from the exact quantum dynamics in the weak system‑reservoir coupling regime without introducing the hypothesis of equilibrium between the system and the reservoir from the beginning; (2) the emergence of thermodynamics in the intermediate system‑reservoir coupling regime where the Born‑Markovian approximation is broken down; (3) the breakdown of thermodynamics due to the long-time non- Markovian memory efect arisen from the occurrence of localized bound states; (4) the existence of dynamical quantum phase transition characterized by infationary dynamics associated with negative dynamical temperature. The corresponding dynamical criticality provides a border separating classical and quantum worlds. The infationary dynamics may also relate to the origin of big bang and universe infation. And the third law of thermodynamics, allocated in the deep quantum realm, is naturally proved. In the past decade, many eforts have been devoted to understand how, starting from an isolated quantum system evolving under Hamiltonian dynamics, equilibration and efective thermodynamics emerge at long times1–5.
    [Show full text]
  • What Is Quantum Thermodynamics?
    What is Quantum Thermodynamics? Gian Paolo Beretta Universit`a di Brescia, via Branze 38, 25123 Brescia, Italy What is the physical significance of entropy? What is the physical origin of irreversibility? Do entropy and irreversibility exist only for complex and macroscopic systems? For everyday laboratory physics, the mathematical formalism of Statistical Mechanics (canonical and grand-canonical, Boltzmann, Bose-Einstein and Fermi-Dirac distributions) allows a successful description of the thermodynamic equilibrium properties of matter, including entropy values. How- ever, as already recognized by Schr¨odinger in 1936, Statistical Mechanics is impaired by conceptual ambiguities and logical inconsistencies, both in its explanation of the meaning of entropy and in its implications on the concept of state of a system. An alternative theory has been developed by Gyftopoulos, Hatsopoulos and the present author to eliminate these stumbling conceptual blocks while maintaining the mathematical formalism of ordinary quantum theory, so successful in applications. To resolve both the problem of the meaning of entropy and that of the origin of irreversibility, we have built entropy and irreversibility into the laws of microscopic physics. The result is a theory that has all the necessary features to combine Mechanics and Thermodynamics uniting all the successful results of both theories, eliminating the logical inconsistencies of Statistical Mechanics and the paradoxes on irreversibility, and providing an entirely new perspective on the microscopic origin of irreversibility, nonlinearity (therefore including chaotic behavior) and maximal-entropy-generation non-equilibrium dynamics. In this long introductory paper we discuss the background and formalism of Quantum Thermo- dynamics including its nonlinear equation of motion and the main general results regarding the nonequilibrium irreversible dynamics it entails.
    [Show full text]
  • Similarity Between Quantum Mechanics and Thermodynamics: Entropy, Temperature, and Carnot Cycle
    Similarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle Sumiyoshi Abe1,2,3 and Shinji Okuyama1 1 Department of Physical Engineering, Mie University, Mie 514-8507, Japan 2 Institut Supérieur des Matériaux et Mécaniques Avancés, 44 F. A. Bartholdi, 72000 Le Mans, France 3 Inspire Institute Inc., Alexandria, Virginia 22303, USA Abstract Similarity between quantum mechanics and thermodynamics is discussed. It is found that if the Clausius equality is imposed on the Shannon entropy and the analogue of the quantity of heat, then the value of the Shannon entropy comes to formally coincide with that of the von Neumann entropy of the canonical density matrix, and pure-state quantum mechanics apparently transmutes into quantum thermodynamics. The corresponding quantum Carnot cycle of a simple two-state model of a particle confined in a one-dimensional infinite potential well is studied, and its efficiency is shown to be identical to the classical one. PACS number(s): 03.65.-w, 05.70.-a 1 In their work [1], Bender, Brody, and Meister have developed an interesting discussion about a quantum-mechanical analogue of the Carnot engine. They have treated a two-state model of a single particle confined in a one-dimensional potential well with width L and have considered reversible cycle. Using the “internal” energy, E (L) = ψ H ψ , they define the pressure (i.e., the force, because of the single dimensionality) as f = − d E(L) / d L , where H is the system Hamiltonian and ψ is a quantum state. Then, the analogue of “isothermal” process is defined to be f L = const.
    [Show full text]
  • Frontiers of Quantum and Mesoscopic Thermodynamics 14 - 20 July 2019, Prague, Czech Republic
    Frontiers of Quantum and Mesoscopic Thermodynamics 14 - 20 July 2019, Prague, Czech Republic Under the auspicies of Ing. Miloš Zeman President of the Czech Republic Jaroslav Kubera President of the Senate of the Parliament of the Czech Republic Milan Štˇech Vice-President of the Senate of the Parliament of the Czech Republic Prof. RNDr. Eva Zažímalová, CSc. President of the Czech Academy of Sciences Dominik Cardinal Duka OP Archbishop of Prague Supported by • Committee on Education, Science, Culture, Human Rights and Petitions of the Senate of the Parliament of the Czech Republic • Institute of Physics, the Czech Academy of Sciences • Department of Physics, Texas A&M University, USA • Institute for Theoretical Physics, University of Amsterdam, The Netherlands • College of Engineering and Science, University of Detroit Mercy, USA • Quantum Optics Lab at the BRIC, Baylor University, USA • Institut de Physique Théorique, CEA/CNRS Saclay, France Topics • Non-equilibrium quantum phenomena • Foundations of quantum physics • Quantum measurement, entanglement and coherence • Dissipation, dephasing, noise and decoherence • Many body physics, quantum field theory • Quantum statistical physics and thermodynamics • Quantum optics • Quantum simulations • Physics of quantum information and computing • Topological states of quantum matter, quantum phase transitions • Macroscopic quantum behavior • Cold atoms and molecules, Bose-Einstein condensates • Mesoscopic, nano-electromechanical and nano-optical systems • Biological systems, molecular motors and
    [Show full text]
  • Quantum Thermodynamics: a Dynamical Viewpoint
    Entropy 2013, 15, 2100-2128; doi:10.3390/e15062100 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Review Quantum Thermodynamics: A Dynamical Viewpoint Ronnie Kosloff Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel; E-Mail: [email protected]; Tel.: + 972-26585485; Fax: + 972-26513742 Received: 26 March 2013; in revised form: 21 May 2013 / Accepted: 23 May 2013 / Published: 29 May 2013 Abstract: Quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. The viewpoint advocated is based on the intimate connection of quan- tum thermodynamics with the theory of open quantum systems. Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics. The emergence of the 0-law, I-law, II-law and III-law of thermodynamics from quantum considerations is presented. The emphasis is on consistency between the two theories, which address the same subject from different foundations. We claim that inconsistency is the result of faulty analysis, pointing to flaws in approximations. Keywords: thermodynamics; open quantum systems; heat engines; refrigerators; quantum devices; finite time thermodynamics 1. Introduction Quantum thermodynamics is the study of thermodynamic processes within the context of quantum dynamics. Thermodynamics preceded quantum mechanics; consistency with thermodynamics led to Planck’s law, the dawn of quantum theory. Following the ideas of Planck on black body radiation, Einstein (1905) quantized the electromagnetic field [1]. Quantum thermodynamics is devoted to unraveling the intimate connection between the laws of thermodynamics and their quantum origin, requiring consistency. For many decades, the two theories developed separately. An exception is the study of Scovil et al.[2,3], which showed the equivalence of the Carnot engine [4] with the three level Maser, setting the stage for new developments.
    [Show full text]
  • The Second Laws of Quantum Thermodynamics
    The second laws of quantum thermodynamics Fernando Brandãoa,1, Michał Horodeckib, Nelly Ngc, Jonathan Oppenheimc,d,2, and Stephanie Wehnerc,e aDepartment of Computer Science, University College London, London WC1E 6BT, United Kingdom; bInstytut Fizyki Teoretycznej i Astrofizyki, University of Gdansk, 80-952 Gdansk, Poland; cCentre for Quantum Technologies, National University of Singapore, 117543 Singapore; dDepartment of Physics & Astronomy, University College London, London WC1E 6BT, United Kingdom; and eSchool of Computing, National University of Singapore, 117417 Singapore Edited by Peter W. Shor, Massachusetts Institute of Technology, Cambridge, MA, and approved January 12, 2015 (received for review June 26, 2014) The second law of thermodynamics places constraints on state Here, we show that this is not the case in the microscopic regime, transformations. It applies to systems composed of many particles, and we therefore needs to talk about “how cyclic” a process is when however, we are seeing that one can formulate laws of thermo- stating the second law. We also derive in this work, a zeroth law of dynamics when only a small number of particles are interacting thermodynamics, which is stronger than the ordinary zeroth law. with a heat bath. Is there a second law of thermodynamics in this For thermodynamics at the macroscopic scale, a system in state regime? Here, we find that for processes which are approximately ρ can be transformed into state ρ′ provided that the free energy cyclic, the second law for microscopic systems takes on a different goes down, where the free energy for a state ρ is form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of FðρÞ = hEðρÞi − kTSðρÞ; [1] constraints.
    [Show full text]
  • SN Bose Annual Report 18
    Annual Report 2018 - 2019 CENTRE FOR BASIC SCIENCES 1986 TIONAL 1894-1974 S. N. BOSE NA SATYENDRA NATH BOSE NATIONAL CENTRE FOR BASIC SCIENCES Annual Report 2018 - 2019 Satyendra Nath Bose National Centre For Basic Sciences Publisher Satyendra Nath Bose National Centre For Basic Sciences Design & Print SKG Media 3rd Floor, 24B Shakespeare Sarani Kolkata - 700 017 Phone : 033 4063 3318 E-mail : [email protected] Academic Highlights No. of Publications in refereed journal 169 No. of PhD Degree awarded 23 No. of PhD theses submitted 21 No. Ongoing Project 31 No. of Patent Applied/ Granted 5 + 2 No. of Awards/Recognitions (Student) 11 No. of Awards/Recognitions (Faculty/Scientist) 7 No. of Technology transfer 1 Source: web of science (On 1st April, 2019) Acknowledgement Annual Report of the ‘Satyendra Nath Bose National Centre for Basic Sciences’ is a brief representation of its activities of a financial year. The report reflects research activities, administrative activities, academic progress and achievement of young research scholars, development of infrastructure and facilities, and establishment of network with advanced research groups around the world. It’s 9th time I have been assigned the job of compilation of Annual Report of the Centre. To prepare the Annual Report, all the faculty members and sections of the Centre spent their valuable time to provide respective data. It is a time bound work to be completed within a short span of time. This is the 3rd time the Annual Report is translated and typed in Hindi within the Centre. The Hindi Officer, Sadhana Tiwari has given sincere fatigueless effort to translate the Annual Report in Hindi and library staff - Gurudas Ghosh and Ananya Sarkar typed the Annual Report in Hindi within a very limited time period.
    [Show full text]
  • Energy-Temperature Uncertainty Relation in Quantum Thermodynamics
    ARTICLE DOI: 10.1038/s41467-018-04536-7 OPEN Energy-temperature uncertainty relation in quantum thermodynamics H.J.D. Miller1 & J. Anders1 It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we 1234567890():,; include such interactions and, using quantum estimation theory, derive a generalised ther- modynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system’s state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase- Dyson skew information. We demonstrate that the temperature’s signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers. 1 Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, UK. Correspondence and requests for materials should be addressed to H.J.D.M. (email: [email protected]) or to J.A. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:2203 | DOI: 10.1038/s41467-018-04536-7 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 ohr suggested that there should exist a form of com- estimate, and we illustrate our bound with an example of a Bplementarity between temperature and energy in thermo- damped harmonic oscillator.
    [Show full text]
  • The Gibbs Paradox: Early History and Solutions
    entropy Article The Gibbs Paradox: Early History and Solutions Olivier Darrigol UMR SPHere, CNRS, Université Denis Diderot, 75013 Paris, France; [email protected] Received: 9 April 2018; Accepted: 28 May 2018; Published: 6 June 2018 Abstract: This article is a detailed history of the Gibbs paradox, with philosophical morals. It purports to explain the origins of the paradox, to describe and criticize solutions of the paradox from the early times to the present, to use the history of statistical mechanics as a reservoir of ideas for clarifying foundations and removing prejudices, and to relate the paradox to broad misunderstandings of the nature of physical theory. Keywords: Gibbs paradox; mixing; entropy; irreversibility; thermochemistry 1. Introduction The history of thermodynamics has three famous “paradoxes”: Josiah Willard Gibbs’s mixing paradox of 1876, Josef Loschmidt reversibility paradox of the same year, and Ernst Zermelo’s recurrence paradox of 1896. The second and third revealed contradictions between the law of entropy increase and the properties of the underlying molecular dynamics. They prompted Ludwig Boltzmann to deepen the statistical understanding of thermodynamic irreversibility. The Gibbs paradox—first called a paradox by Pierre Duhem in 1892—denounced a violation of the continuity principle: the mixing entropy of two gases (to be defined in a moment) has the same finite value no matter how small the difference between the two gases, even though common sense requires the mixing entropy to vanish for identical gases (you do not really mix two identical substances). Although this paradox originally belonged to purely macroscopic thermodynamics, Gibbs perceived kinetic-molecular implications and James Clerk Maxwell promptly followed him in this direction.
    [Show full text]